>>12673538>A real number is a number that can be assigned a location on the number line.That doesn't actually mean anything. You are trying to define an arithmetical number system in terms of a geometric object. Therefore, you've just passed the buck of what a real number is to what a "location on the number line" is. And while there are many subtle issues of what this exactly means (as
>>12673910 mentions), the more obvious issue is that of avoiding circularity.
How are you going to define a line? If you stick with the usual delusion that a line is an infinite set of points, then what are points? Points today are defined in terms of a number from some base number system (rationals if you are practical and "reals" if you are fanciful). So if you don't change anything else other than your definition of reals, then you're essentially saying "real numbers are locations on the number line and a location on the number is a point specified by a real number." In other words, you haven't actually gotten anywhere.
And, if you leave a point as your undefined object, you're essentially taking the same approach to mathematics that Euclid had 2000 years ago. Introducing coordinates to geometry is really powerful since a lot of Euclid's postulates could now be theorems, since we can precisely define points, lines, etc. in terms of numbers and equations. If you abandon that, you basically lose everything from the 16-17th century onwards in terms of mathematics.
For example, how do you know whether two curves intersect? Today, you'd check to see if there is a point at which they meet, but you haven't actually defined a point so you're basically taking Euclid's approach of just drawing things somewhere and eyeballing.
Therefore, you basically have to "define" a real "number" arithmetically and not geometrically to avoid circularity and maintain a coordinate-based geometry.
In summary, accept your fate and bow down to the cult of the Berger.
